### abstract

- A Bohr-Sommerfeld equation is derived for the highly damped quasinormal mode frequencies $\ensuremath{\omega}(n\ensuremath{\gg}1)$ of rotating black holes. It may be written as $2{\ensuremath{\int}}_{C}({p}_{r}+i{p}_{0})dr=(n+1/2)h$, where ${p}_{r}$ is the canonical momentum conjugate to the radial coordinate $r$ along a null geodesic of energy $\ensuremath{\hbar}\ensuremath{\omega}$ and angular momentum $\ensuremath{\hbar}m$, ${p}_{0}=O({\ensuremath{\omega}}^{0})$, and the contour $C$ connects two complex turning points of ${p}_{r}$. The solutions are $\ensuremath{\omega}(n)=\ensuremath{-}m\stackrel{^}{\ensuremath{\omega}}\ensuremath{-}i(\stackrel{^}{\ensuremath{\phi}}+n\stackrel{^}{\ensuremath{\delta}})$, where ${\stackrel{^}{\ensuremath{\omega}},\stackrel{^}{\ensuremath{\delta}}}>0$ are functions of the black-hole parameters alone. Some physical implications are discussed.